The original definition of the amplituhedron in [1]} is purely geometrical: the amplituhedron is the image in \({\rm Gr}_{k,k+m}\)  of the nonnegative Grassmannian \({\rm Gr}_{k,n}^{\ge 0}\)  by a map. This definition has the advantage to be simple but it is very hard to check whether or not a point of \({\rm Gr}_{k,k+m}\)  belongs to the amplituhedron. In [2]}, Arkani-Hamed, Trnka and Thomas gave two conjectural definitions of the amplituhedron, one for \(m\)  even and the other one for \(m\)  odd, which answer this question. Depending on the parity of \(m\) , they associate an integer to each point of \({\rm Gr}_{k,k+m}\)  called the winding number or the crossing number. To compute these numbers it suffices to calculate the so-called twistor coordinates, themselves obtained by computings determinants of matrices. Then, they conjectured that a point of \({\rm Gr}_{k,k+m}\)  is inside \(\mathcal {A}_{n,k,m}\)  if and only if it satisfies some boundary conditions and the crossing or winding number has a specific value depending on \(k\)  and \(m\) . The boundary conditions are also checked by computing twistor coordinates, thus, according to their definition, it suffices to compute determinants to know if a point of \({\rm Gr}_{k,k+m}\)  is inside the amplituhedron or not.
